according to a direct application of Jensen's inequality (cf. Theorem 8.8 in [1]} and the discussion therein), where \(\Vert d\Vert _\infty \)  is a bound on the maximal distance. These results indicate that the posterior may not contract at a rate faster than the optimal convergence rate of point estimators. However, little is known in the literature about the converse statement in the nonparametric setting, that is, whether the seemingly simpler convergence rate of Bayes estimators can lead to the rate of convergence of the entire posterior distribution.
